关于首导数处含小参数的偏积分微分方程的差分估计

A new class of mild and strong solutions of integro-differential equation of arbitrary order in Banach space

The motivation that the field of differential equations provide to several researchers for the challenges that have been challenging them over the decades has contributed to the strengthening of the area within mathematics. In this sense, investigating important properties of solutions of differential equations, in particular fractional, has been object of study due to the exponential growth of the fractional calculus. In this paper, we investigate the existence and uniqueness of a new class of mild and strong solution of the fractional integro-differential equations in the Hilfer fractional d…

Analysis of Hilfer fractional integro-differential equations with almost sectorial operators

We investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove existence results by applying Schauder's fixed point technique. Moreover, we show fundamental properties of the solution representation by discussing two cases related to the associated semigroup. For that, we consider compactness and noncompactness properties, respectively. Furthermore, an example is given to illustrate the obtained theory.

具有非对称正核的完全非线性积分-微分算子的正则性结果：次临界情形

Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels : Subcritical Case

我们引入了一类新的完全非线性积分-微分算子可能的非对称内核，其中包括所产生的随机控制问题的纯跳Lèvy过程。若算子$σ$的指数在$ (1,2)$中（次临界情形），则我们得到了与这类算子相关的完全非线性积分微分方程的比较原理、Alexandroff-Backelman-Pucci估计的非局部形式、Harnack不等式、Hölder正则性和内$\rm C^{1,α}$ -正则性.而且，当算子的指数$σ$接近2时，我们的估计仍然是一致的，因此，我们的估计可以看作是椭圆型偏微分方程正则性结果的自然扩张.

KdV方程的Airy核行列式解与积分微分Painlevé方程

Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations

本文研究由变形Airy核Fredholm行列式对数导数构造的KdV方程无界解，揭示其与积分微分Painlevé方程的联系，并给出t→0时的一致渐近，改进KPZ方程窄楔解尾部估计。

Some inequalities for partial derivatives on time scales

We first prove some weighted inequalities for compositions of functions on time scales which are in turn applied to establish some new dynamic Opial-type inequalities in several variables. Some generalizations and applications to partial differential dynamic equations are also considered.

Existence Results for a Multipoint Fractional Boundary Value Problem in the Fractional Derivative Banach Space

We study a class of nonlinear implicit fractional differential equations subject to nonlocal boundary conditions expressed in terms of nonlinear integro-differential equations. Using the Krasnosel'skii fixed point theorem we prove, via the Kolmogorov--Riesz criteria, existence of solutions. The existence results are established in a specific fractional derivative Banach space and they are illustrated by two numerical examples.

Integrated solutions of non-densely defined semilinear integro-differential inclusions: existence, topology and applications

Given a linear closed but not necessarily densely defined operator $A$ on a Banach space $E$ with nonempty resolvent set and a multivalued map $F\colon I\times E\map E$ with weakly sequentially closed graph, we consider the integro-differential inclusion \begin{center} $\dot{u}\in Au+F(t,\int u)\;\;\text{on }I,\;\;u(0)=x_0.$ \end{center} We focus on the case when $A$ generates an integrated semigroup and obtain existence of integrated solutions in the sense of \cite[Def.6.4.]{thieme} if $E$ is weakly compactly generated and $F$ satisfies \[β(F(t,Ω))\<η(t)β(Ω)\;\;\text{for all bounded }Ω\subset…

带记忆项的常微分方程的希尔伯特空间透视

A Hilbert Space Perspective on Ordinary Differential Equations with Memory Term

讨论了希尔伯特空间中具有时滞和记忆项的常微分方程。通过在适当的希尔伯特空间中引入一个时间导数作为正规算子，我们发展了一种新的方法，在一个统一的框架内研究了积分微分方程、中立型微分方程和一般时滞微分方程的解理论.我们表明，合理的微分方程导致因果解算子。

Explicit solutions for linear variable-coefficient fractional differential equations with respect to functions

Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite series of fractional integro-differential operators, which can be widely and efficiently used for analytic and computational purposes. In the case of constant coefficients, the solution can be expressed in terms of the multivariate Mittag-Leffler functions. In particular, the obtained result extends the Luchko-Gorenflo representation formula to a general clas…

On a problem of Mazur from "The Scottish Book" concerning second partial derivatives

We comment on a Mazur problem from "Scottish Book" concerning second partial derivatives. It is proved that, if a function $f(x,y)$ of real variables defined on a rectangle has continuous derivative with respect to $y$ and for almost all $y$ the function $\,F_y(x):=f'_y(x,y)$ has finite variation, then almost everywhere on the rectangle there exists the partial derivative $f"_{yx}$. We construct a separately twice differentiable function, whose partial derivative $f'_x$ is discontinuous with respect to the second variable on a set of positive measure. This solves in the negative the Mazur prob…

参数导数$[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=0}$和$[\partial^{3}P_ν(z)/\partialν^{3}]_{ν=0}$，其中$P_ν(z)$是第一类勒让德函数

The parameter derivatives $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=0}$ and $[\partial^{3}P_ν(z)/\partialν^{3}]_{ν=0}$, where $P_ν(z)$ is the Legendre function of the first kind

我们推导出明确的表达式的参数导数$[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=0}$和$[\partial^{3}P_ν(z)/\partialν^{3}]_{ν=0}$，其中$P_ν(z)$是第一类的勒让德函数。我们发现{displaymath} \frac{\partial^{2}P_ν(z)}{\partialν^{2}} \bigg|_{ν=0} =-2 \Li_{2} \frac{1-z}{2}，{displaystyle}其中$\Li_{2}z$是双对数（Schramkowski最近使用\emph{Mathematica}得出了此公式），{displaystyle} \frac{\partial^{3}P_ν(z)}{\partialν^{3}} \bigg|_{v =0} =12 \Li_{3} \frac{z+1}{2} -6 \ln \frac{z+1}{2} \Li_{2} \frac{z+1}{2} -π^{2} \ln \frac{z+1}{2} -12 <$（3），{displaystyle}其中$\Li_{3}z$是3阶的多重对数，$ζ(s)$是黎曼zeta函数。